This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator is Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.

Dans cette thèse, nous présentons des méthodes asymptotiques <br />mathématiquement justifiées permettant de connaître les champs <br />électromagnétiques dans des milieux à couches minces hétérogènes. <br />La motivation de ce travail est le calcul du champ électrique dans des <br />cellules biologiques composées d'un cytoplasme conducteur entouré <br />d'une fine membrane très isolante. <br />Nous remplaçons la membrane, lorsque son épaisseur est infiniment <br />petite, par des conditions de transmission ou des conditions aux <br />limites appropriées et nous estimons l'erreur commise par ces <br />approximations.<br /> Pour les basses fréquences, nous considérons l'équation quasistatique<br />donnant le potentiel dont dérive le champ. A l'aide d'un <br />calcul en géométrie circulaire nous obtenons les expressions explicites<br /> du potentiel et nous en déduisons les asymptotiques du champ <br />électrique, en fonction de l'épaisseur de la couche mince, avec des <br />estimations de l'erreur. Nous estimons ensuite la différence entre le <br />champ réel et le champ statique. Puis nous généralisons notre <br />développement asymptotique à une géométrie quelconque. <br /> La deuxième partie de cette thèse traite des moyennes fréquences : <br />nous donnons le développement asymptotique de la solution de <br />l'équation de Helmholtz lorsque l'épaisseur de la membrane tend vers <br />0. Tous ces précédents résultats sont illustrés par des calculs par <br />éléments finis.<br /> Enfin, pour les hautes fréquences, nous construisons une condition <br />d'impédance pseudodifférentielle permettant de concentrer l'effet de <br />la couche sur son bord intérieur. Nous concluons cette thèse par un <br />problème de diffraction à haute fréquence d'une onde incidente par <br />un disque de petite taille. A l'aide d'une analyse pseudodifférentielle, <br />nous bornons la norme de la trace du champ diffracté à distance fixe <br />de l'inhomogénéité en fonction de la taille de l'objet et de l'onde <br />incidente.

Presented at Cyber-Enabled Discovery and Innovation: Knowledge Extraction as a success story lecture. See for more detail https://www.ipam.ucla.edu/programs/cdi2007/

This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

No description available.

Doctor of Philosophy / Department of Mathematics / Virginia Naibo / Leibniz-type rules associated to bilinear pseudodifferential operators have received considerable attention due to their applications in obtaining fractional Leibniz rules and the study of various partial differential equations. Generally speaking, fractional Leibniz rules provide a way of estimating the size and smoothness of a product of functions in terms of the size and smoothness of the individual functions themselves. Such rules are helpful in determining well-posedness results for solutions of PDEs modeling a variety of real world phenomena, ranging from Euler and Navier-Stokes equations (which model incompressible fluid flow, such as airflow over a wing) to Korteweg-de Vries equations (which model waves on shallow water surfaces). Bilinear pseudodifferential operators act to combine two functions using their Fourier transforms and a symbol, which is a function that assigns different weights to the functions’ frequency components as they are combined. Thus, Leibniz-type rules associated to bilinear pseudodifferential operators serve as a generalization of fractional Leibniz rules by providing estimates on the size and smoothness of some combination of two functions, for which pointwise multiplication is recoverable by choosing a symbol identically equal to one. A variety of function spaces may be used to measure the size and smoothness of functions involved, including Lebesgue spaces, Sobolev spaces, and Besov and Triebel-Lizorkin spaces. Further, bilinear pseudodifferential operators may be considered in association with different classes of symbols, which is to say that the symbol itself (and possibly its derivatives) will possess certain decay properties. New Leibniz-type rules in two different settings will be presented in this manuscript. In the first setting, Leibniz-type rules associated to bilinear pseudodifferential operators with homogeneous symbols in a certain class are proved, where the sizes of the functions involved are measured using a combination of Lebesgue space norms and norms corresponding to function spaces admitting appropriate molecular decompositions, specifically focusing on the case of homogeneous Besov-type and Triebel-Lizorkin-type spaces. In the second setting, Leibniz-type rules and biparameter counterparts are proved in weighted Lebesgue and Sobolev spaces associated to Coifman-Meyer multiplier operators. All of the new Leibniz-type rules proved in the manuscript yield corresponding new fractional Leibniz rules, which are highlighted as appropriate. Various techniques from Fourier analysis serve as important tools in the proofs of these new results, such as obtaining paraproduct decompositions for bilinear pseudodifferential operators and utilizing Littlewood-Paley theory and square function-type estimates.

Dans cette thèse nous avons étudié, d'une part le théorème d'absorption limitepour des opérateurs de Schrödinger et de Dirac avec des potentiels oscillants. Lefait de considérer des potentiels oscillants est intéressant dans la mesure où ses opé-rateurs peuvent avoir des valeurs propres plongées dans le spectre continu (c'est lecas pour Schrödinger), ce qui est plutôt inhabituel et introduit de nouvelles di-cultés. L'étude du théorème d'absorption limite est très importante pour la théoriede la diffusion. Un intérêt particulier du sujet réside dans le fait que l'outil naturelpour procéder à l'étude en question, à savoir la théorie du commutateur de Mourre,ne s'applique pas. Une alternative récente a été développée par les co-directeurs dela thèse Thierry Jecko et Sylvain Golénia. Elle a été appliquée à un opérateur deSchrödinger avec potentiel oscillant. Il s'agit donc d'améliorer les résultats sur lesopérateurs de Schrödinger et de traiter le cas des opérateurs de Dirac. D'autre part,nous avons montré un résultat de type Helffer-Sjöstrand pour les opérateurs unitaires.Et pour finir, nous avons pu montrer l'existence des valeurs propre plongéespour l'opérateur de Dirac avec des potentiels relativement compact par rapport àl'opérateur de Dirac libre sur son spectre essentiel. / In this thesis, we have studied the limit absorption theorem for Schrödinger andDirac operators with oscillating potentials. Considering oscillating potentials is interestinginsofar as its operators can have of the eigenvalues plunged into the continuousspectrum (this is the case for Schrödinger), which is rather unusual and introducesnew dificulties. The study of the limit absorption theorem is very important for thetheory of diffusion. A particular interest of the subject lies in the fact that the naturaltool for the study in question, namely the Mourre switch theory, does not apply. Arecent alternative has been developed by the co-directors Thierry Jecko and SylvainGolénia. It has been applied to a Schrödinger operator with oscillating potential. Itis therefore a question of improving the results on the Schrödinger operators and oftreating the case of Dirac operators. Secondly, we have shown a Helffer-Sjöstrandformula for the unit operators and finally we have been able to show the existenceof the eigenvalues plunged for the Dirac operator with relatively compact potentialsrelative to the operator of free Dirac on its essential spectrum.

Nesta tese vamos apresentar dois resultados a respeito de K-teoria de álgebras C^{*} de classes de operadores pseudodiferenciais que são globalmente definidos em \\mathbb^. O primeiro resultado é a prova da regularidade da função \\eta para operadores clássicos com símbolos de Shubin. Vamos mostrar que a álgebra de operadores pseudodiferenciais em \\mathbb^ com símbolos de Shubin permite a construção de potências complexas e um tipo de traço de Kontsevich-Vishik numa forma muito similar àquela feita para variedades compactas, com definições até mais simples. Mostraremos, então, que podemos definir as funções \\zeta e \\eta também para esses símbolos. Finalmente mostraremos como o conhecimento de fatos simples sobre a sua K-teoria permitem a prova da regularidade da função \\eta. Para variedades compactas, esse resultado tem muitas implicações. Acreditamos assim que ele também possa ser interessante para os estudos de operadores globais em \\mathbb^. O segundo resultado é o cálculo da K-teoria de operadores limitados gerados por operadores de Boutet de Monvel SG de ordem (0,0) e tipo zero em \\mathbb_{+}^. Boutet de Monvel introduziu a álgebra que leva o seu nome para estudar o índice de operadores elípticos de fronteira em variedades compactas com bordo. Mais recentemente uma nova abordagem foi proposta por Melo, Nest, Schrohe e Schick para obter resultados sobre o índice de Fredholm usando a K-teoria de álgebras C^{*}, uma ferramenta que não era disponível ainda quando Boutet de Monvel desenvolveu sua álgebra. Nossa ideia foi, então, mostrar como calcular a K-teoria de álgebras de Boutet de Monvel com símbolos SG em \\mathbb_{+}^, em que os símbolos SG são uma classe de símbolos globalmente definidos em \\mathbb^. Acreditamos que isso possa ser útil também ao estudo de problemas elípticos de fronteira para operadores de Boutet de Monvel com símbolos SG em certas classes de variedades não compactas. / We are going to present two results concerning K-theory of C^{*} algebras of classes of pseudodifferential operators that are globally defined in \\mathbb^. The first result is the proof of the regularity of the \\eta function for classical operators with Shubin symbols. We are going to show that the algebra of classical pseudodifferential operators in \\mathbb^ with Shubin symbols allows the construction of complex powers and a kind of Kontsevich-Vishik trace in a very similar way as on compact manifolds, with even easier definitions. Then we show that we can define the \\zeta and \\eta functions also for these symbols. Finally we will show how the knowledge of simple facts about the K-theory of pseudodifferential operators with Shubin\'s symbols allows the proof of the regularity of the \\eta function at 0. For compact manifolds, this regularity is a result that has many implications. Therefore it may also be interesting for global operators in \\mathbb^. The second result is the evaluation of the K-theory of bounded operators generated by SG Boutet de Monvel operators of order (0,0) and type 0 in \\mathbb_^. Boutet de Monvel introduced his algebra to study the index of elliptic boundary value problems on compact manifolds. More recently a new approach was proposed by Melo, Nest, Schrohe and Schick to obtain results about the index of Fredholm operators using the K-theory of C^ algebras, a tool which was not well known when Boutet de Monvel published his work. The idea here is to show how one can evaluate the K-theory of the Boutet de Monvel operators with SG symbols in \\mathbb_^, where SG symbols is a class of symbols globally defined in \\mathbb^. We believe that this can be useful to the study of index of Fredholm problems also in the case of Boutet de Monvel operators with SG symbols in some classes of non-compact manifolds.

Nesta tese vamos apresentar dois resultados a respeito de K-teoria de álgebras C^{*} de classes de operadores pseudodiferenciais que são globalmente definidos em \\mathbb^. O primeiro resultado é a prova da regularidade da função \\eta para operadores clássicos com símbolos de Shubin. Vamos mostrar que a álgebra de operadores pseudodiferenciais em \\mathbb^ com símbolos de Shubin permite a construção de potências complexas e um tipo de traço de Kontsevich-Vishik numa forma muito similar àquela feita para variedades compactas, com definições até mais simples. Mostraremos, então, que podemos definir as funções \\zeta e \\eta também para esses símbolos. Finalmente mostraremos como o conhecimento de fatos simples sobre a sua K-teoria permitem a prova da regularidade da função \\eta. Para variedades compactas, esse resultado tem muitas implicações. Acreditamos assim que ele também possa ser interessante para os estudos de operadores globais em \\mathbb^. O segundo resultado é o cálculo da K-teoria de operadores limitados gerados por operadores de Boutet de Monvel SG de ordem (0,0) e tipo zero em \\mathbb_{+}^. Boutet de Monvel introduziu a álgebra que leva o seu nome para estudar o índice de operadores elípticos de fronteira em variedades compactas com bordo. Mais recentemente uma nova abordagem foi proposta por Melo, Nest, Schrohe e Schick para obter resultados sobre o índice de Fredholm usando a K-teoria de álgebras C^{*}, uma ferramenta que não era disponível ainda quando Boutet de Monvel desenvolveu sua álgebra. Nossa ideia foi, então, mostrar como calcular a K-teoria de álgebras de Boutet de Monvel com símbolos SG em \\mathbb_{+}^, em que os símbolos SG são uma classe de símbolos globalmente definidos em \\mathbb^. Acreditamos que isso possa ser útil também ao estudo de problemas elípticos de fronteira para operadores de Boutet de Monvel com símbolos SG em certas classes de variedades não compactas. / We are going to present two results concerning K-theory of C^{*} algebras of classes of pseudodifferential operators that are globally defined in \\mathbb^. The first result is the proof of the regularity of the \\eta function for classical operators with Shubin symbols. We are going to show that the algebra of classical pseudodifferential operators in \\mathbb^ with Shubin symbols allows the construction of complex powers and a kind of Kontsevich-Vishik trace in a very similar way as on compact manifolds, with even easier definitions. Then we show that we can define the \\zeta and \\eta functions also for these symbols. Finally we will show how the knowledge of simple facts about the K-theory of pseudodifferential operators with Shubin\'s symbols allows the proof of the regularity of the \\eta function at 0. For compact manifolds, this regularity is a result that has many implications. Therefore it may also be interesting for global operators in \\mathbb^. The second result is the evaluation of the K-theory of bounded operators generated by SG Boutet de Monvel operators of order (0,0) and type 0 in \\mathbb_^. Boutet de Monvel introduced his algebra to study the index of elliptic boundary value problems on compact manifolds. More recently a new approach was proposed by Melo, Nest, Schrohe and Schick to obtain results about the index of Fredholm operators using the K-theory of C^ algebras, a tool which was not well known when Boutet de Monvel published his work. The idea here is to show how one can evaluate the K-theory of the Boutet de Monvel operators with SG symbols in \\mathbb_^, where SG symbols is a class of symbols globally defined in \\mathbb^. We believe that this can be useful to the study of index of Fredholm problems also in the case of Boutet de Monvel operators with SG symbols in some classes of non-compact manifolds.

The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.